Asymptotic properties of the growth curve model with covariance components

We consider a multivariate regression (growth curve) model of the form , where and are unknown scalar covariance components. In the case of replicated observations, we derive the explicit form of the locally best estimators of the covariance components under normality and asymptotic confidence ellipsoids for certain linear functions of the first order parameters {B _ij} estimating simultaneously the first and the second order parameters.     

Integration over a fractal curve and the jump problem

A definition of integration, i.e., a generalization of a functional of the form

Probability properties and fractal properties of statistically recursive sets

In this paper we construct a class of statistically recursive sets K by statistical contraction operators and prove the convergence and the measurability of K. Many important sets are the special cases of K. Then we investigate the statistically self-similar measure (or set). We have found some sufficient conditions to ensure the statistically recursive set to be statistically self-similar. We also investigate the distribution PK-1. The zero-one laws and the support of PK-1 are obtained.Finally the Hausdorff dimension and Hausdorff exact measure function of a class of statistically recursive sets constructed by a collection of i.i.d. statistical contraction operators have been obtained.     

The growth curve model with an autoregressive covariance structure

The growth curve model with an autoregressive covariance structure is considered. An iterative algorithm for finding the MLE's of the parameters in the model is presented, based on the modified likelihood equations. Asymptotic distributions of the MLE's are obtained when the sample size is large. A likelihood ratio statistic for testing the autoregressive covariance structure is presented.     

Asymptotic properties of nonparametric curve estimates

We consider a class of nonparametric estimators for the regression functionm(t) in the model:y _i =m(t _i ) + _i , 1 i n, t _i [0, 1], which are linear in the observationsy _i . Several limit theorems concerning local and global stochastic and a.s. convergence and limit distributions are given.     

The Fibonacci fractal is a new fractal type

We propose a uniform method for estimating fractal characteristics of systems satisfying some type of scaling principle. This method is based on representing such systems as generating Bethe-Cayley tree graphs. These graphs appear from the formalism of the group bundle of Fibonacci-Penrose inverse semigroups. We consistently consider the standard schemes of Cantor and Koch in the new method. We prove the fractal property of the proper Fibonacci system, which has neither a negative nor a positive redundancy type. We illustrate the Fibonacci fractal by an original procedure and in the coordinate representation. The golden ratio and specific inversion property intrinsic to the Fibonacci system underlie the Fibonacci fractal. This property is reflected in the structure of the Fibonacci generator.     

Residuals in the growth curve model

Residuals for the Growth Curve model will be discussed. In univariate linear models as well as the ordinary multivariate analysis of variance model residuals are based on the difference between the observations and the mean whereas for the Growth Curve model we have three different residuals all showing various aspects useful for validating analysis. For these residuals some basic properties are established.     

Estimating common parameters of growth curve models

Suppose that we have two independent random matrices X ₁ and X ₂ having multivariate normal distributions with common unknown matrix of parameters (q×m) and different unknown covariance matrices ₁ and ₂, given by N _{p1, N1} (B ₁ A ₁; ₁, I) and N _{p2, N2} (B ₂ A ₂; ₂, I) respectively. Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] (% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\]) be the MLE of based on X ₁ (X ₂) only. When q=1, necessary and sufficient conditions that a combined estimator of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] has uniformly smaller covariance matrix than those of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] are given. The k-sample problem as well as one-sample problem is also discussed. These results are extensions of those of Graybill and Deal (1959, Biometrics, 15, 543–550), Bhattacharya (1980, Ann. Statist., 8, 205–211; 1984, Ann. Inst. Statist. Math., 36, 129–134) to multivariate case.Dedicated to Professor Yukihiro Kodama on his 60th birthday.Bowling Green State UniversityVisiting Professor on leave from the University of Tsukuba, Japan. Now at Department of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan.This research was partially supported by University of Tsukuba Project Research 1986.     

Generalized fractal dimensions: equivalences and basic properties

Given a positive probability Borel measure μ on , we establish some basic properties of the associated functions τ_μ^±(q) and of the generalized fractal dimensions D_μ^±(q) for . We first give the equivalence of the Hentschel–Procaccia dimensions with the Rényi dimensions and the mean-q dimensions, for q>0. We then use these relations to prove some regularity properties for τ_μ^±(q) and D_μ^±(q); we also provide some estimates for these functions, in particular estimates on their behaviour at ±∞, as well as for the dimensions corresponding to convolution of two measures. We finally present some calculations for specific examples illustrating the different cases met in the article.     

Thermal properties of bodies in fractal and cantorian physics

Fundamental laws describing the heat diffusion in fractal environment are discussed. It is shown that for the three-dimensional space the heat radiation process occur in structures with fractal dimension D 0,1), whereas in structures with D (1,3 heat conduction and convection have the upper hand (generally in the real gases).To describe the heat diffusion a new law has been formulated. Its validity is more general than the Plank’s radiation law based on the quantum heat diffusion theory. The energy density w = f (K, D), where K is the fractal measure and D is the fractal dimension exhibit typical dependency peaking with agreement with Planck’s radiation law and with the experimental data for the absolutely black body in the energy interval kT < Kc. The positions of the energy density maximums (for fractal dimensions D_m < 0.31854) are in a good agreement with the maximums determined by Wien’s displacement law with the help of the Lambert’s W- Function u(A) = A + W[−Aexp(−A)], where A ≈ 1.9510 and u = hc/λ_mkT_m ≈ 1.5275. The agreement of the fractal model with the experimental outcomes is documented for the spectral characteristics of the Sun. The properties of stellar objects (black holes, relict radiation, etc.) and the elementary particles fields and interactions between them (quarks, leptons, mesons, baryons, bosons and their coupling constants) will be discussed with the help of the described mathematic apparatus in our further contributions.The general gas law for real gases in its more applicable form than the widely used laws (e.g. van der Waals, Berthelot, Kammerlingh–Onnes) has been also formulated. The energy density, which is in this case represented by the gas pressure p = f (K, D), can gain generally complex value and represents the behaviour of real (cohesive) gas in interval D (1,3. The gas behaves as the ideal one only for particular values of the fractal dimensions (the energy density is real-valued). Again, it is shown that above the critical temperature (kT > Kc) and for fractal dimension D_m > 2.0269 the results are comparable to the kinetics theory of real (ideal) gas (van der Waals equation of state, compressibility factor, Boyle’s temperature). For the critical temperature (Kc = kT_r) the compressibility factor gains Z = 1 (except for the ideal gas case D = 3) also for the fractal dimension D = 1/ = 1.618033989, where is the golden mean value of the El Naschie’s golden mean field theory. To determine the minimum it is also possible to employ the Lambert’s W− Function u(A) = A + W[−Aexp(−A)], whereA ≈ 0.6779 and u ≈ −0.7330. The thermal properties of fractal structures (thermal capacity, thermal conductivity, diffusivity) and additional parameters (enthalpy, entropy, etc.) will be defined using the mathematic apparatus in the future. Good agreement of the fractal model with experimental data is documented on the compressibility factor of various gases.     

Some properties of fractal interpolation functions on Sierpinski gasket

In this paper, we discuss some basic properties of uniform fractal interpolation functions (FIFs), which is a special class of FIFs, on Sierpinski gasket. We firstly study the min-max property of uniform FIFs. Then we present a necessary and sufficient condition such that uniform FIFs have finite energy. Normal derivative and Laplacian of uniform FIFs are also discussed.     

On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μξ of the following random variables: where ak are terms of a given positive convergent series; ξk are independent random variables taking values 0 and 1 with probabilities p0k and p1k correspondingly.We give (without any restriction on {an}) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μξ itself are studied in details for the case where ak?rk:=ak+1+ak+2+? (i.e., rk−1?2rk) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μξ for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results.     

Isochronous properties in fractal analysis of some planar vector fields

We extend the classical Belitskii normal form theorem uniformly for planar nilpotent foci and limit cycles. Then by using fractal analysis, isochronous properties near such invariant sets are well investigated.     

Linear sufficiency in a general growth curve model

The notion of linear sufficiency for the whole set of estimable functions in the general Gauss-Markov model is extended to the estimation of any special set of estimable functions in a general growth curve model. Some general results with respect to the concept of linear sufficiency are obtained, from which a necessary and sufficient condition is established for a linear transformation, {F₁,F₂}, of the observation matrix Y to have the property that there exists a linear function of which is the BLUE of the estimable functions .     

Special variance structures in the growth curve model

Standard and extended growth curve model (multivariate linear model) with practically important variance structures are considered and a method for parameters estimation is proposed.     

Topological and fractal properties of real numbers which are not normal

The set L of essentially non-normal numbers of the unit interval (i.e., the set of real numbers having no asymptotic frequencies of all digits in their nonterminating s-adic expansion) is studied in details. It is proven that the set L is generic in the topological sense (it is of the second Baire category) as well as in the sense of fractal geometry (L is a superfractal set, i.e., the Hausdorff-Besicovitch dimension of the set L is equal 1). These results are substantial generalizations of the previous results of the two latter authors [M. Pratsiovytyi, G. Torbin, Ukrainian Math. J. 47 (7) (1995) 971-975].The Q^∗-representation of real numbers (which is a generalization of the s-adic expansion) is also studied. This representation is determined by the stochastic matrix Q^∗. We prove the existence of such a Q^∗-representation that almost all (in the sense of Lebesgue measure) real numbers have no asymptotic frequency of all digits. In the case where the matrix Q^∗ has additional asymptotic properties, the Hausdorff-Besicovitch dimension of the set of numbers with prescribed asymptotic properties of their digits is determined (this is a generalization of the Eggleston-Besicovitch theorem). The connections between the notions of “normality of numbers” respectively of “asymptotic frequencies” of their digits is also studied.     

Testing some covariance structures under a growth curve model

This paper considers three types of problems: (i) the problem of independence of two sets, (ii) the problem of sphericity of the covariance matrix Σ, and (iii) the problem of intraclass model for the covariance matrix Σ, when the column vectors of X are independently distributed as multivariate normal with covariance matrix Σ and E(X) = BξA,A and B being given matrices and ξ and Σ being unknown. These problems are solved by the likelihood ratio test procedures under some restrictions on the models, and the null distributions of the test statistics are established.     

On some tests of growth curve model under Behrens-Fisher stituation

In this article, we have considered the problem of testing equality of several growth curves under Behrens-Fisher situation. In this context, the robustness of the existing test criteria have been studied. Also, some exact test procedures have been considered and the exact and asymptotic noncentral distribution problems have been studied.